[j«y]
(material equivalence; the biconditional)

[j«y]
has the following truth-table:

jy

[j«y]

T
T

T

T
F

F

F
T

F

F
F

T

One
can read it as "j double
arrow y"; or as "j
if and only ify".

Using it
one can translate "John likes rugby if and only if he is Welsh"
as "[John likes rugby « John is Welsh]". When translating "if and only if"
with "[j«y]" one needs,
of course, to exercise the same caution as when translating "if"
with "[j®y]".

Notice that
we already had a way of translating "j
if and only if y" without
using "[j«y]". We could have
paraphrased it as "j if
y, and j
only if y", and translated
that as "[[y®j]Ù[j®y]]".
The translations come to the same thing. "[j«y]" is false just
when "j" and "y"
have different truth-values. "[[y®j]Ù[j®y]]"
is false just when "[y®j]"
is false or "[j®y]"
is false; that is, it is false just when either "y"
is true and "j" is
false or when "y" is
true and "j" is false;
that is, just when "j"
and "y" have different
truth-values.

This
is a rather a wordy way of arguing. One could represent the same facts
neatly by using a truth-table:

jy

[j«y]

[[y®j]

Ù

[j®y]]

T
T

T

T

T

T

T
F

F

T

F

F

F
T

F

F

F

T

F
F

T

T

T

T

From this
it can be seen that the two translations will be true in exactly the same
circumstances. (To draw a truth table, one works from the less complicated
expressions to the more complicated expressions; one writes the truth-values
under the corresponding symbol. So the right hand column of values represents
the values that "[j®y]"
takes in each case; and the column to its left represents the values that
"[[y®j]Ù[j®y]]"
takes; and so on.)

We noticed
earlier that one way of translating "j
OR y", where the "or"
is intended to be exclusive would be to use "[[jÚy]Ù¬[jÙy]]".
A neat way of saying the same thing would be "[j«¬y]".

Let us use
a truth-table to check this.

jy

[j«

¬y]

[[yÚj]

Ù

¬

[jÙy]]

T
T

F

F

T

F

F

T

T
F

T

T

T

T

T

F

F
T

T

F

T

T

T

F

F
F

F

T

F

F

T

F

From this
we can see that both are true just when the constituents take different
truth-values.